Related papers: A Basis-preserving Algorithm for Computing the Bez…
Computing a basis for the exponent lattice of algebraic numbers is a basic problem in the field of computational number theory with applications to many other areas. The main cost of a well-known algorithm…
We use Newton's method to find all roots of several polynomials in one complex variable of degree up to and exceeding one million and show that the method, applied to appropriately chosen starting points, can be turned into an algorithm…
We combine the known methods for univariate polynomial root-finding and for computations in the Frobenius matrix algebra with our novel techniques to advance numerical solution of a univariate polynomial equation, and in particular…
Constructive methods for matrices of multihomogeneous (or multigraded) resultants for unmixed systems have been studied by Weyman, Zelevinsky, Sturmfels, Dickenstein and Emiris. We generalize these constructions to mixed systems, whose…
An algorithm and its first implementation in C# are presented for assembling arbitrary quantum circuits on the base of Hadamard and Toffoli gates and for constructing multivariate polynomial systems over the finite field Z_2 arising when…
In this paper, we make a contribution to the computation of Gr\"obner bases. For polynomial reduction, instead of choosing the leading monomial of a polynomial as the monomial with respect to which the reduction process is carried out, we…
We show a new algorithm and its implementation for multiplying bit-polynomials of large degrees. The algorithm is based on evaluating polynomials at a specific set comprising a natural set for evaluation with additive FFT and a high order…
Univariate polynomial root-finding is a classical subject, still important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the…
We consider systems of recursively defined combinatorial structures. We give algorithms checking that these systems are well founded, computing generating series and providing numerical values. Our framework is an articulation of the…
An improved characteristic set algorithm for solving Boolean polynomial systems is proposed. This algorithm is based on the idea of converting all the polynomials into monic ones by zero decomposition, and using additions to obtain…
In this paper we present an algorithm for construction of minimal involutive polynomial bases which are Groebner bases of the special form. The most general involutive algorithms are based on the concept of involutive monomial division…
For systems of polynomial equations, we study the problem of computing the Newton polytope of their eliminants. As was shown by Esterov and Khovanskii, such Newton polytopes are mixed fiber polytopes of the Newton polytopes of the input…
Classically, the Bezout matrix or simply Bezoutian of two polynomials is used to locate the roots of the polynomial and, in particular, test for stability. In this paper, we develop the theory of Bezoutians on real Riemann surfaces of…
In numerical simulations of many charged systems at the micro/nano scale, a common theme is the repeated solution of the Poisson-Boltzmann equation. This task proves challenging, if not entirely infeasible, largely due to the nonlinearity…
Quasi-Newton methods are well known techniques for large-scale numerical optimization. They use an approximation of the Hessian in optimization problems or the Jacobian in system of nonlinear equations. In the Interior Point context,…
This paper is a detailed description of an algorithm based on a generalized Buchberger algorithm for constructing Groebner-type bases associated with polynomials of shift operators. The algorithm is used for calculating Feynman integrals…
An algorithm to compute a good basis of the Brieskorn lattice of a cohomologically tame polynomial is described. This algorithm is based on the results of C. Sabbah and generalizes the algorithm by A. Douai for convenient Newton…
When exploring the literature, it can be observed that the operator obtained when applying \textit{Newton-like} root finding algorithms to the quadratic polynomials $z^2-c$ has the same form regardless of which algorithm has been used. In…
An algorithm for matrix factorization of polynomials was proposed in \cite{fomatati2022tensor} and it was shown that this algorithm produces better results than the standard method for factoring polynomials on the class of summand-reducible…
The classical persistence algorithm computes the unique decomposition of a persistence module implicitly given by an input simplicial filtration. Based on matrix reduction, this algorithm is a cornerstone of the emergent area of topological…